# A study of Universal ODE approaches to predicting soil organic carbon

arXiv: https://arxiv.org/abs/2509.24306 · HTML: https://mlsanity.com/paper/2509.24306/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 69 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Builds on (most cited first, 10 of 10)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEs](https://mlsanity.com/paper/2404.10024/) (2024)
- [A Comprehensive Survey of Regression Based Loss Functions for Time Series Forecasting](https://mlsanity.com/paper/2211.02989/) (2022)
- [Predicting and Mapping of Soil Organic Carbon Using Machine Learning Algorithms in Northern Iran](https://mlsanity.com/paper/2007.12475/) (2020)
- [SSL-SoilNet: A Hybrid Transformer-based Framework with Self-Supervised Learning for Large-scale Soil Organic Carbon Prediction](https://mlsanity.com/paper/2308.03586/) (2023)

## Abstract (arXiv)

Soil Organic Carbon (SOC) is a foundation of soil health and global climate resilience, yet its prediction remains difficult because of intricate physical, chemical, and biological processes. In this study, we explore a Scientific Machine Learning (SciML) framework built on Universal Differential Equations (UDEs) to forecast SOC dynamics across soil depth and time. UDEs blend mechanistic physics, such as advection diffusion transport, with neural networks that learn nonlinear microbial production and respiration. Using synthetic datasets, we systematically evaluated six experimental cases, progressing from clean, noise free benchmarks to stress tests with high (35%) multiplicative, spatially correlated noise. Our results highlight both the potential and limitations of the approach. In noise free and moderate noise settings, the UDE accurately reconstructed SOC dynamics. In clean terminal profile at 50 years (Case 4) achieved near perfect fidelity, with MSE = 1.6e-5, and R2 = 0.9999. Case 5, with 7% noise, remained robust (MSE = 3.4e-6, R2 = 0.99998), capturing depth wise SOC trends while tolerating realistic measurement uncertainty. In contrast, Case 3 (35% noise at t = 0) showed clear evidence of overfitting: the model reproduced noisy inputs with high accuracy but lost generalization against the clean truth (R2 = 0.94). Case 6 (35% noise at t = 50) collapsed toward overly smooth mean profiles, failing to capture depth wise variability and yielding negative R2, underscoring the limits of standard training under severe uncertainty. These findings suggest that UDEs are well suited for scalable, noise tolerant SOC forecasting, though advancing toward field deployment will require noise aware loss functions, probabilistic modelling, and tighter integration of microbial dynamics.